Backward stochastic Volterra integral equations on Markov chains

Abstract

This paper studies Backward Stochastic Volterra Integral Equations (BSVIEs) driven by finite state, continuous time Markov chains. First, the existence and uniqueness of the solutions to two types of BSVIEs are established. Second, some scalar and vector comparison theorems are given. Finally, the applications of BSVIEs to a linear-quadratic optimal control problem and time-inconsistent coherent risk measures are presented.

Keywords:

• Backward stochastic Volterra integral equation
• Markov chains
• comparison theorem
• dynamic risk measures

Acknowledgements

The author would like to thank two anonymous referees for their valuable comments and suggestions to improve on earlier versions of this paper, and thank Dr Tianxiao Wang for valuable discussions on BSVIEs.

Notes

No potential conflict of interest was reported by the author.

1 The arguments in that paper still hold for the BSVIEs driven by Markov chains.

Additional information

Funding

This work was sponsored by Program of Shanghai Subject Chief Scientist [grant number 14XD1401600]; the 111 Project [grant number B14019]; National Natural Science Foundation of China [grant number 11231005], [grant number 11571113], [grant number 11601157].